Moderate deviations principle for empirical covariance from a unit root

In the present paper, we consider the linear autoregressive model in $\rr$, $X_{k,n}=θ_n X_{k,n-1}+ξ_k, k=0,1,...,n, n\ge 1$ where $θ_n\in [0,1)$ is unknown, $(ξ_k)_{k\in\zz}$ is a sequence of centered i.i.d. r.v. valued in $\rr$ representing the noise. When $θ_n\to 1$, the moderate deviations principle for empirical covariance is discussed and as statistical applications we provide the moderate deviation estimates of the least square and the Yule-Walker estimators of the parameter $θ_n$.

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